Does blender recognize the same background if present in multiple frames of an animation?
At least in cycles, no, and in other renderers, very unlikely.

In order to get physically accurate light transport you can't make assumptions like this, and Cycles definitely tries to be accurate in this regard.

Think about it, where does the computer decide where to cut from background to middle to foreground. How does it deal with reflected light or shadow onto the background? The only way to accurately figure this out is to trace the light rays, at which point you are rendering it. 

If you want a static background, you render it separately and either composite it, use an emit only background pane or environment texture. This is generally the preferable solution as it is both simpler from a developer support perspective, and also puts control squarely in the artist's hands.